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Find the sum of the infinite geometric series.\newline3+32+34+38+…3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_

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Q. Find the sum of the infinite geometric series.\newline3+32+34+38+…3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term aa and the common ratio rr of the series. The first term is the first number in the series, and the common ratio is the factor by which each term is multiplied to get the next term.\newlineIn this series, the first term aa is 33, and the common ratio rr can be found by dividing the second term by the first term, or the third term by the second term, and so on.\newlineCalculating the common ratio: r=(32)/3=12r = (\frac{3}{2}) / 3 = \frac{1}{2}
  2. Calculate Common Ratio: The sum SS of an infinite geometric series with ∣r∣<1|r| < 1 is given by the formula S=a1−rS = \frac{a}{1 - r}, where aa is the first term and rr is the common ratio.\newlinePlugging in the values we have: S=31−12S = \frac{3}{1 - \frac{1}{2}}
  3. Calculate Sum: Now we perform the calculation for the sum SS.\newlineS=31−12=312=3×21=6S = \frac{3}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 3 \times \frac{2}{1} = 6\newlineThe sum of the infinite geometric series is 66.

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